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<style>.article-header[data-astro-cid-t54qzdpc]{margin:4em 0}.article-header[data-astro-cid-t54qzdpc] h1[data-astro-cid-t54qzdpc].article-title{font-size:2.2em;text-align:center;font-weight:500;margin:0 var(--gap-1)}.article-header[data-astro-cid-t54qzdpc] .article-subtitle[data-astro-cid-t54qzdpc]{color:var(--font-subtitle-color);text-align:center;margin:var(--gap-1)}.article-header[data-astro-cid-t54qzdpc] .article-subtitle[data-astro-cid-t54qzdpc] a[data-astro-cid-t54qzdpc]{color:unset}.article-header[data-astro-cid-t54qzdpc] .article-date[data-astro-cid-t54qzdpc]{color:var(--font-subtitle-color);text-align:center;margin:var(--gap-1) 0;font-style:italic}
.admonition{--admonition-color-bg: rgb(218, 248, 255);--admonition-color-fg: rgb(35, 173, 204);--admonition-color-text: var(--font-normal-color)}.admonition{padding:var(--gap-0);background-color:var(--admonition-color-bg);border:1px dotted var(--admonition-color-fg);border-radius:.2em;color:var(--admonition-color-text)}
.todo-block{padding:1em;background:repeating-linear-gradient(45deg,#f9f7ff,#f9f7ff 10px,#eee9ff 10px 20px);border:2px solid #d8ccff}
.demo canvas{display:block;max-width:100%;margin:0 auto}
</style></head> <body> <header id="header-box" data-astro-cid-3uvkrtzm> <nav id="header" data-astro-cid-3uvkrtzm> <ul id="header-nav" data-astro-cid-3uvkrtzm> <li data-astro-cid-3uvkrtzm> <a id="header-name" href="/" data-astro-cid-3uvkrtzm>Benjamin R. Bray</a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/blog" class="active" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Blog </a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/projects" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Projects </a> </li> <!-- <li class="header-button"><HeaderLink href="/series">Series</a></li> --> <!-- <li class="header-button"><HeaderLink href="/favorites">Favorites</a></li> --> <li class="header-button" data-astro-cid-3uvkrtzm><a href="https://github.com/benrbray" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> GitHub </a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/resume-pdf" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Resume </a> </li> <!-- <li class="header-button"><HeaderLink href="/sitemap">Sitemap</a></li> --> </ul> </nav> </header> <main> <div class="column-thin" data-astro-cid-lnnucpux> <article> <div class="article"> <!-- draft --><div class="article-header" data-astro-cid-t54qzdpc> <!-- title --> <h1 class="article-title" data-astro-cid-t54qzdpc>Abstract Algebra: Modules</h1> <!-- subtitle --> <div class="article-subtitle" data-astro-cid-t54qzdpc> </div> <!-- date --> <div class="article-date" data-astro-cid-t54qzdpc>
Published <time datetime="2020-05-13T00:00:00.000Z"> May 13, 2020 </time> </div> </div> <div class="article-body"> <span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow></mrow><annotation encoding="application/x-tex">\providecommand{\FF}{\mathbb{F}}
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<!DOCTYPE html><html lang="en"> <head><!-- Global Metadata --><meta charset="utf-8"><meta name="viewport" content="width=device-width,initial-scale=1"><link rel="icon" type="image/svg+xml" href="/favicon.ico"><!-- Font preloads --><link rel="preload" href="/fonts/lora/Lora-VariableFont_wght.ttf" as="font" type="font/ttf" crossorigin><link rel="preload" href="/fonts/lora/Lora-Italic-VariableFont_wght.ttf" as="font" type="font/ttf" crossorigin><!-- Canonical URL --><link rel="canonical" href="https://benrbray.com/blog/"><!-- Primary Meta Tags --><title>Blog Posts / Benjamin R. Bray</title><meta name="title" content="Blog Posts / Benjamin R. Bray"><link rel="stylesheet" href="/_astro/_slug_.YvsiLoxm.css">
<style>.preview-link[data-astro-cid-gaprsfch]{display:block;color:unset;text-decoration:unset}.preview-box[data-astro-cid-gaprsfch]{margin:var(--gap-1) 0;padding:var(--gap-1);border-radius:var(--gap-0);background-color:#f6f6f6;border:1px dashed #ccc}.preview-box[data-astro-cid-gaprsfch]:hover{background-color:#eee}.preview-title[data-astro-cid-gaprsfch]{font-size:var(--font-size-large);font-weight:var(--font-title-weight);text-decoration:2px limegreen underline}.preview-title[data-astro-cid-gaprsfch] a[data-astro-cid-gaprsfch]{color:unset}.preview-summary[data-astro-cid-gaprsfch]{margin-top:var(--gap-0)}
.demo canvas{display:block;max-width:100%;margin:0 auto}
</style></head> <body> <header id="header-box" data-astro-cid-3uvkrtzm> <nav id="header" data-astro-cid-3uvkrtzm> <ul id="header-nav" data-astro-cid-3uvkrtzm> <li data-astro-cid-3uvkrtzm> <a id="header-name" href="/" data-astro-cid-3uvkrtzm>Benjamin R. Bray</a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/blog" class="active" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Blog </a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/projects" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Projects </a> </li> <!-- <li class="header-button"><HeaderLink href="/series">Series</a></li> --> <!-- <li class="header-button"><HeaderLink href="/favorites">Favorites</a></li> --> <li class="header-button" data-astro-cid-3uvkrtzm><a href="https://github.com/benrbray" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> GitHub </a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/resume-pdf" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Resume </a> </li> <!-- <li class="header-button"><HeaderLink href="/sitemap">Sitemap</a></li> --> </ul> </nav> </header> <main> <div class="column-normal"> <main> <h1>Blog Posts</h1> <section> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/blog/rust/rust-wasm" data-astro-cid-gaprsfch>Minimal Example of Rust&#8209;Wasm with WebGL and <code>specs</code></a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2024-07-31T15:00:00.000Z"> Aug 1, 2024 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>A minimal Demo page for Rust-Wasm</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/blog/abstract-algebra-modules" data-astro-cid-gaprsfch>Abstract Algebra: Modules</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2020-05-13T00:00:00.000Z"> May 13, 2020 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>Vector spaces over a field are a special case of the more general notion of modules over a ring. Rather than the long list of axioms normally presented in textbooks, we see how an algebraic view of vector spaces helps to motivate the definition of modules.</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/hilbert-spaces_5aug19.pdf" data-astro-cid-gaprsfch>Notes: Hilbert Spaces & Adjoints</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2019-08-05T00:00:00.000Z"> Aug 5, 2019 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>TODO</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/matrix-inversion-lemma_12jul19.pdf" data-astro-cid-gaprsfch>Notes: Matrix Inversion Lemma</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2019-07-12T00:00:00.000Z"> Jul 12, 2019 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>TODO</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/topology-summary_jun18_draft.pdf" data-astro-cid-gaprsfch>Notes: Topology Basics</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2018-06-21T00:00:00.000Z"> Jun 21, 2018 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>A summary of basic concepts in topology, including topological spaces, compactness, separation, convergence, and continuity.</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/caratheodory-radon-helly_jun17.pdf" data-astro-cid-gaprsfch>Notes: Carathéodory, Radon, and Helly Theorems</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2017-06-21T00:00:00.000Z"> Jun 21, 2017 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>TODO</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/separating-hyperplane-thm_jun17.pdf" data-astro-cid-gaprsfch>Notes: Separating & Supporting Hyperplanes</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2017-06-06T00:00:00.000Z"> Jun 6, 2017 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>TODO</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/sup-inf-inequalities_jun17.pdf" data-astro-cid-gaprsfch>Notes: Sup-Inf Inequalities</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2017-06-05T00:00:00.000Z"> Jun 5, 2017 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>TODO</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/chebychev-polynomials_dec16.pdf" data-astro-cid-gaprsfch>Notes: Chebychev Polynomials</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2016-12-08T00:00:00.000Z"> Dec 8, 2016 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>The <b>Chebyshev polynomials</b> appear frequently in numerical analysis and are incredibly useful for analyzing and accelerating the convergence of iterative methods. One might even say that Chebyshev polynomials are the best polynomials, a fact which can be made precise in a variety of different ways. In these notes, we define Chebyshev polynomials and their basic properties, before discussing their utility in minimax approximation theory, which was the subject of a previous set of notes.</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/minimax-approx_nov16.pdf" data-astro-cid-gaprsfch>Notes: Minimax Approximation Theory</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2016-11-27T00:00:00.000Z"> Nov 27, 2016 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>First formalized by Weierstrass in 1885, approximation theory concerns the best approximation of arbitrary functions by some class of simpler functions. The structure of an approximation problem involves three central components: a function class containing the function to be approximated, a form of approximating function, and a norm for measuring approximation error. These notes introduce <b>minimax polynomial approximation</b>, whereby continuous functions on a closed interval are approximated by polynomials using the infinity norm to measure fit.</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> <div class="preview-box" data-astro-cid-gaprsfch> <div class="preview-title" data-astro-cid-gaprsfch> <a href="/notes/dynkin-pi-lambda_nov16.pdf" data-astro-cid-gaprsfch>Notes: Dynkin's Pi-Lambda Theorem</a> <span style="float:right; color: gray; font-weight: normal; font-size: var(--font-normal-size)" data-astro-cid-gaprsfch> <time datetime="2016-11-15T00:00:00.000Z"> Nov 15, 2016 </time> </span> </div> <div class="preview-summary" data-astro-cid-gaprsfch>TODO</div> </div> <!-- consider this method for block links --> <!-- https://css-tricks.com/block-links-the-search-for-a-perfect-solution/#method-4-sprinkle-javascript-on-the-second-method --> </section> </main> </div> </main> <footer class="footer" data-astro-cid-x4grat5q> <div class="footer-website" data-astro-cid-x4grat5q>
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<style>.article-header[data-astro-cid-t54qzdpc]{margin:4em 0}.article-header[data-astro-cid-t54qzdpc] h1[data-astro-cid-t54qzdpc].article-title{font-size:2.2em;text-align:center;font-weight:500;margin:0 var(--gap-1)}.article-header[data-astro-cid-t54qzdpc] .article-subtitle[data-astro-cid-t54qzdpc]{color:var(--font-subtitle-color);text-align:center;margin:var(--gap-1)}.article-header[data-astro-cid-t54qzdpc] .article-subtitle[data-astro-cid-t54qzdpc] a[data-astro-cid-t54qzdpc]{color:unset}.article-header[data-astro-cid-t54qzdpc] .article-date[data-astro-cid-t54qzdpc]{color:var(--font-subtitle-color);text-align:center;margin:var(--gap-1) 0;font-style:italic}
.admonition{--admonition-color-bg: rgb(218, 248, 255);--admonition-color-fg: rgb(35, 173, 204);--admonition-color-text: var(--font-normal-color)}.admonition{padding:var(--gap-0);background-color:var(--admonition-color-bg);border:1px dotted var(--admonition-color-fg);border-radius:.2em;color:var(--admonition-color-text)}
.todo-block{padding:1em;background:repeating-linear-gradient(45deg,#f9f7ff,#f9f7ff 10px,#eee9ff 10px 20px);border:2px solid #d8ccff}
.demo canvas{display:block;max-width:100%;margin:0 auto}
</style></head> <body> <header id="header-box" data-astro-cid-3uvkrtzm> <nav id="header" data-astro-cid-3uvkrtzm> <ul id="header-nav" data-astro-cid-3uvkrtzm> <li data-astro-cid-3uvkrtzm> <a id="header-name" href="/" data-astro-cid-3uvkrtzm>Benjamin R. Bray</a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/blog" class="active" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Blog </a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/projects" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Projects </a> </li> <!-- <li class="header-button"><HeaderLink href="/series">Series</a></li> --> <!-- <li class="header-button"><HeaderLink href="/favorites">Favorites</a></li> --> <li class="header-button" data-astro-cid-3uvkrtzm><a href="https://github.com/benrbray" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> GitHub </a> </li> <li class="header-button" data-astro-cid-3uvkrtzm><a href="/resume-pdf" data-astro-cid-3uvkrtzm data-astro-cid-oujquvsc> Resume </a> </li> <!-- <li class="header-button"><HeaderLink href="/sitemap">Sitemap</a></li> --> </ul> </nav> </header> <main> <div class="column-thin" data-astro-cid-lnnucpux> <article> <div class="article"> <!-- draft --><div class="article-header" data-astro-cid-t54qzdpc> <!-- title --> <h1 class="article-title" data-astro-cid-t54qzdpc>Notes: Carathéodory, Radon, and Helly Theorems</h1> <!-- subtitle --> <div class="article-subtitle" data-astro-cid-t54qzdpc> </div> <!-- date --> <div class="article-date" data-astro-cid-t54qzdpc>
Published <time datetime="2017-06-21T00:00:00.000Z"> Jun 21, 2017 </time> </div> </div> <div class="article-body"> </div> </div> </article> </div> </main> <footer class="footer" data-astro-cid-x4grat5q> <div class="footer-website" data-astro-cid-x4grat5q>
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